We hope that this response reaches you in the best of health and Imaan. Ameen inshaAllah.
It is ok so say Alhamdulillah around non-Muslims. However, it is best to refrain from saying "Assalaamu Alaikum" or "Jazak Allah" to them because both are in essence duas you bestow on Muslims. One dua you can make for a non-Muslim is a dua that Allah Ta'ala grants them guidance and hidayah.
In order to show good manners (akhlaaq), though, one may greet a non-Muslim with a kind dua such as "May Allah grant you righteousness and guidance."
And instead of saying "Jazak Allah", you may use words such as "Thank you" to show your gratitude.
And Allah Knows best
And Allah knows best.
Ask Alimah Team
Note: The rulings giving herein are based on the religious rulings of the Islamic Law and do not have any implications on the Law of the Country. The rulings given hereunder are specifically based on the question posed and should be read in conjunction with the question. AskAlimah.com site bears no responsibility to any party who may or may not act on this answer. AskAlimah.com site being hereby exempted from loss or damage howsoever caused. This answers may not be used as evidence in any Court of Law without prior written consent of the web site.