I've noticed that sometimes when I am writing out equations and saying the symbols in my head I will slip up and say "p" for "$ \rho $" or mix the pronunciation between letters say "$ \eta $" for "$ \xi $" possibly because the latter symbol looks like a cursive "e" where the former sounds like the pronunciation of "e". I was worried that I was practicing bad habits and so decided to try to learn the alphabet with the correct pronunciation for use in mathematics once and for all.
Modern Greek pronunciation differs significantly from how I've heard the alphabet pronounced and so I think it is clear that living in the United States I should follow the American English pronunciation. However, there still seem to be considerable differences in pronunciation, or perhaps I am just obsessing over it. For example, two different American English pronunciations of zeta:
http://www.merriam-webster.com/dictionary/zeta
http://www.oxfordlearnersdictionaries.com/definition/english/zeta?q=zeta