I agree with Patricia that "correlation is not causation" - so just because 2 things happened at the same time (or one shortly after the other) doesn't mean that one caused the other.
I also question the decision to give antibiotics for the flu. Antibiotics are totally ineffective against a virus. Is it possible that you didn't understand why they were being given? Unless they had a raging sinus or ear infection at the same time, there would be no need for antibiotics, and in fact this would be a bad idea (antibiotics are overused and give rise to super-bugs). Did they perhaps need an inhaler because they also had breathing problems? Just throwing out suggestions here.
I understand completely that you were worried about your 2 month old - but instead of seeking out a second opinion, you chose not to take her in anywhere until 9 months. So the pediatrician rightly concluded that you are not in agreement with the philosophy and guidelines of the practice, and you would be better off with someone else. You say that "the people who I had entrusted the well-being of my children instead of helping me and guiding me gave me the boot." But you didn't trust them really, did you? You disagree with pretty much everything they told you, you insist that your children are reacting and becoming ill because of what they told you to do, right? You say they told you to leave because you complained about the vaccines - but that's not the reason. The real reason is, you refused to bring your infant in for anything, from 2 months to 9 months of age. So you should want to move on even more than they want you to leave. Many, many pediatricians are refusing to see patients who do not agree with vaccines - it's not safe for their other patients (including very young babies) to have other people in the waiting room who are not vaccinated. That's how these deadly diseases spread. And it's how diseases that were virtually eliminated from the U.S. have now become active again.
Also, you don't believe the doctor's diagnosis of esotropia - so you have to find a doctor who will partner with you and let you do some of the diagnosing. I don't know the best way to do that. The AAPOS, the American experts in children's eye issues, says nothing about this condition being related to antibiotics. http://www.aapos.org/terms/conditions/48
If you are insisting that vaccines cause problems, you must find an alternative physician who agrees with you that these are harmful. You may also find someone who will do an alternative schedule with you (spreading out vaccines vs. clustering them together). You could also find out a lot more about the ingredients in the vaccines from different manufacturers - preservatives and so forth. Sometimes the vaccine is the same but the method of "storage" is different. There was a question about this not long ago on Mamapedia - you can search for it and read the answers.
I think you might benefit from taking your children to a major children's hospital for a full battery of tests and diagnoses that you will believe. A second opinion is always a good idea, and one major medical center that houses all the specialties you need would ensure that your children's care was fully coordinated. If you want referrals in your area, I suggest you post a new question, click the button on the question page that lets you restrict it to the region you're in, and also put the geographic region right in the title of your question. I also think you should take a friend with you - someone who is not closely involved, who is objective, who is not nervous in medical situations, and who has a good command of English especially in medical settings, and who will take good notes for you. That's always a good idea in situations where a parent is overwhelmed with so much info, so many worries, and one or more children to deal with in the appointment room.
And no, you will probably not be able to enroll any unvaccinated children in public school without vaccinations, unless you have documentation from a qualified, board-certified physician that your children suffer from some autoimmune disease that makes vaccines impossible for them. You can home school, of course, but you still should not take your children into situations where they will expose others to the diseases, or where they will be exposed to other people (adults and children) who refused vaccinations. So that means that parks, stores, children's museums, kids' classes and activities, and any crowded location is a risk for them and others. It may also include hospitals - so you will have to work out something with any children's hospital to even have them allow your unvaccinated child in for those tests and evaluations I mentioned above.
I hope you get the information you seek.