No. Your tires locking is based on the static friction of the tires to the road.
Once the static friction is overcome, regardless of whether it is from engine braking or normal braking, the tires will lock.
I'll expand this answer based on the edits.
Your tires locking is based on static friction between your tire and the road. Braking on ice will lock the tires much easier than wet asphalt, which is also easier than dry asphalt.
Your ability to brake is based on your brakes and engine braking, and other source of friction applied to the rotation of your tires. You can also do other (highly NOT recommended stuff) like put your gear in reverse, go into gear and shut down your engine, etc. Whatever it is that causes the wheels to not rotate as well.
Where the source of braking comes from don't matter. You simply want to get as close to the static friction limit as possible without exceeding the static friction limit.