Sacked for being on my period and putting head on desk at call centre and answering calls everytime it rings...?

In America, every state is 'at-will' - meaning an employer can let an employee go for no reason and an employee can quit a job for no reason.  On your next job.....do your job and do NOT allow your monthly issues keep you from doing your job.

You find a new job. And if your period affects you so badly that you can’t function normally, you see your doctor. Your behavior was unprofessional, regardless of the reason, and your employer does not owe you another chance, especially as you were on probation.

Start looking for another job.  You were let go because you put your head down and rested during you shift.  Because it is still a probationary period, the company doesn't have to have much of a reason to terminate you.  I'd say you're lucky.  Do you really want to work with or for a group of people who would report you rather than ask you if you were unwell? Did no one at least mention to you not to rest your head?  Obviously, you should know that laying your head on your desk isn't appropriate or professional -- and you should have spoken to your supervisor if you didn't think you could perform you duties adequately.  Life and learn.

As an aside, if you normally can't function during your period, see an ob/gyn.  You can't expect to take time of every month or to rest on the job. 

You haven't said which country you're in so I'll just answer as if you were in the same country as myself.

The problem is you were on probation which means the employer didn't need a good reason to fire you and you have no recourse in such event.  You are still protected by the Equality Act regardless how long you've been employed and so if you advised the employer at the start of the shift that you were suffering due to being on your period and they still sacked you you might be able to bring a claim for discrimination and unfair dismissal.  The thing is it will be very hard to prove as the employer could simply say that your performance was below par and as a probationary they decided to let you go.  If they employ other females in the same role who have passed their probabation then your claim will almost certainly fail.

The best thing to do would be to look for another job and also see if you can take some prescribed medication that will cause you not to have your period during your probation.  Alternatively, look for a job where you can work from home.

30 minutes out of 8 hours is huge. You are not paid to sleep.

You were not sacked for being on your period, but for putting your head on your desk, a behavior your former employer did not appreciate.