If there were a simple and obvious answer to this riddle, such as low brake fluid or a gross brake fluid leak, one can assume that any mechanic who has been in business more than 15 minutes would have figured it out. Let's dig a little deeper.
I think your mechanic messed up and won't admit it. (Or, more charitably, in his ignorance he doesn't realize that his work caused your brake failure.)
Brake pedal height adjustment is a touchy, risky operation on a Civic. If you lower the brake pedal too much, it causes a very slight drag between the calipers and brake discs. The drag can be too slight to detect, but it causes the brake parts to heat up. The hot caliper causes the fluid to expand, and because the pedal has been adjusted too low, the brake fluid can not return freely to the master cylinder.
This creates more and more drag, more heat, and eventually the brake fluid in one or more calipers can vaporize.
There's a brake safety monitor to detect any pressure difference between the two halves of the brake system. If a pressure difference is detected, the monitor turns on the dashboard BRAKES light. This is the same light that comes on when you have low fluid or severely worn brake pads.
I think what happened when you stepped on the brakes is that the caliper with vaporized brake fluid took a full pedal's worth of fluid and developed no pressure, while the other half of your brake system did develop brake pressure, but not enough to effectively slow down. The difference would have turned on your BRAKES light. I think you could have developed effective stopping power by pumping your brakes -- a lost art since ABS systems were developed, so you'll be forgiven for never learning the technique.
You didn't mention it, but there may have been a burning odor when you pulled over and one or more of your wheels was probably very hot. You may have missed this if the hot brake was in the rear and/or there was a good headwind and you never got downwind of the hot brake.
You are still at risk of losing your brakes again unless the mechanic has surreptitiously re-adjusted your brake pedal height properly.
You need a new mechanic.
The brake pedal height needs to be checked and re-adjusted by someone experienced in the art, not a random mechanic who has never done it before.
Your braking system needs to be thoroughly inspected to look for glazed / burned / damaged parts that resulted from overheating.
Your brake fluid should be flushed and replaced with new fluid if there is evidence that overheating occurred.
EDIT AFTER EXCHANGING COMMENTS
Based on your original question, I assumed that you went to the mechanic because you were not happy with the height of the brake pedal. Some people have trouble moving their foot from the gas pedal to the brake pedal because the brake pedal is so much higher, and they can catch their instep on the back of the brake pedal. It is possible to carefully lower the brake pedal on a Civic by adjusting a nut on the master cylinder's shaft under the dash and another nut on the brake light switch in the same area.
But your comments now reveal that this is not why you originally went to the mechanic.
Your complaint was that the brake pedal travel that was needed to begin engaging the brakes was greater than you are accustomed to -- not the brake pedal height. The mechanic made the assumption that the rear drum brakes were out of adjustment, so he adjusted the rear brakes to bring the brake shoes closer to the brake drums so that less pedal travel would be needed before the brakes engage. Your comments reveal that he did not adjust the pedal itself, i.e., he did not adjust the nut on the master cylinder shaft under the dash. You assumed that he adjusted the brake pedal, but he only adjusted the rear drum brakes.
Disc brakes never need adjusting; they are self-adjusting every time you step on the brake pedal. Drum brakes are adjusted when first installed, and then as they wear, a mechanism checks for any needed adjustment whenever you step on the brake while backing up. If adjustment is needed, they are automatically made tighter by a single "click" of a rotating star wheel or ratchet wheel in the brake mechanism, and that suffices for many weeks or months. It's not enough of an adjustment to notice any difference in pedal height.
It's possible for the self-adjustment mechanism to get rusty and to stop adjusting. If this happens, as your brake drums and brake shoes wear, you'll notice that the brake pedal travels farther and farther before the brakes start braking because it takes more brake fluid to move the brake shoes into contact with the brake drums.
The proper procedure when this happens is not to simply adjust the brakes, but to clean and lubricate or totally replace the rusted brake adjustment mechanism. If parts are available, it's an easy repair. The parts are not expensive. When the repair is finished, the most prudent way to adjust the rear brakes is to adjust most of the way to where the brake shoes need to be, then take the car out into the parking lot, drive backwards and step on the brake repeatedly until the brakes self-adjust into the factory-designed clearance.
It appears that your mechanic didn't do this, but instead adjusted the rear brakes until they were dragging slightly. This guaranteed that your brake pedal would travel only a short distance before the brakes would engage.
Unfortunately, he maladjusted the brakes to have too much drag. At this point, the story coincides with my original narrative of what happened. Hot brakes, worse and worse drag until the brake parts were so hot that the brake fluid boiled. You lost braking ability because so much fluid was used to re-fill the hot rear brake cylinder(s) when you stepped on the brake.
Your rear brake adjustment mechanism must be either de-rusted and lubricated or replaced. Then the rear brakes must be properly adjusted, allowing the self-adjusting mechanism to make the final, working adjustment. When complete, your brake pedal will require a certain amount of travel before engaging. This is by design. If you previously had 4-wheel disc brakes and now you have drum brakes in the rear, this accounts for the different feel. You need to adjust to it, not the other way around. When the automatic adjusting mechanism is new or clean and lubricated, the brake pedal travel will be as designed at the factory.