Am I legally allowed to work 10 hour days without a break?

This is probably illegal. You should be entitled to a meal break, and not "on the clock". You should even be allowed to leave the premises and eat wherever and whatever you want. Usually you are entitled to at least half an hour.

Unfortunately, if you squawk about this, you may get fired. It's difficult and expensive to fight unfair labor practices. Maybe other posters will have useful suggestions.

In my state (in USA), one is guaranteed a 15 minute (paid) break for every four hours of work and is allowed a 30 minute meal break during a 6 hour shift (paid or unpaid, at the employer's discretion).

It depends what country you are in. You need to check the labour laws where you are. I very much doubt that what your employer is doing is legal. Where I am, you have to have a half hour break after four hours of work and another break after eight hours. Any work over eight hours is time and a half.

But it all depends where you are. Look it up online.

in the U.S. that is legal. your employer is not required to give you breaks, although OSHA guidelines say you must be allowed "reasonable use" of a restroom, though it does not go into detail so it's subject to interpretation.

In Australia that isn't legal. What a shitty boss, even if it's legal it is morally wrong.