She is not a breed. A breed has many specific requirements: size, comb type, earlobe color, egg shell color, skin color, body shape, number of toes, and many many others. If you have to put a label on her you can call her a mixed breed, mutt, barnyard mix, or cross breed. She is really a chicken. If you are happy with her you are happy with her and that's what counts.
The way I tell whether a hen is truly broody and deserves eggs is where she spends the nights. If she spends two consecutive nights on the nest instead of sleeping in her normal spot I consider her worthy of eggs.
You marked the eggs and have been through a broody before so it sounds like you know what you are doing.
I'll attach a thread where someone had a broody hen in the cold of Michigan instead of your fairly mild climate in Texas. It's the best documented case of winter brooding I've seen.
http://www.backyardchickens.com/t/947046/broody-in-michigan-winter
Before a hen even starts to lay she builds up excess fat. That excess fat is what a broody mostly lives on when she is on the nest. That way she can take care of the eggs instead of needing to be off the nest looking for food all the time. Yes she loses weight as she uses that excess fat but it is fat put there for that very reason. Her health is not threatened because she uses fat put there for that very reason.
