As a general matter, employment in the state of Texas is "at will" absent an express agreement to the contrary. At will employment can be terminated for any reason not amounting to discrimination on the basis of a legally protected trait (race, religion, gender, etc.) or retaliation for engaging in certain forms of legally protected conduct (filing a wage claim, taking FMLA leave, etc.). It doesn't matter whether the basis for termination is fair, reasonable or even true.
This means that you can generally be terminated for something that wasn't your fault. It also means that your employer doesn't need you to sign a writeup for them to terminate you for the conduct they allege in the write up.
What complicates matters a little bit is the HIPAA issue. HIPAA has what's called a "Breach Notification Rule" (45 CFR §§ 164.400-414), which requires HIPAA covered entities to provide notification following a breach of unsecured protected health information. If this trainee made copies of protected health information in the binder or otherwise disseminated such information, your employer would be obligated to inform the patients. However, to be honest, it is not clear from these facts whether HIPPA was actually violated as there is no proof that copies of protected health information were made or of any other violation of the Act. It could be assumed, but a speculated HIPAA violation is probably not enough to trigger the notification requirement. There would need to be some concrete evidence of an actual violation.
Now, that said, you could certainly report this incident to the Department of Health & Human Services, which is the entity that enforces HIPAA. Making this report would b e a form of legally protected conduct, which as explained above means that you could not be terminated in retaliation for it. The choice to make such a complaint would be yours.
If I can clarify anything at all for you, please do not hesitate to ask. It is my pleasure to assist you further if necessary....