The detect browser requirement by the bank's website is to check if you have compatible browser for their security features. And for the secure connection question, always use secure connection when accessing something as private as your bank account online. Do not compromise on security.
"Detecting browser requirements" means that the website is verifying if you have a secure web browser (offering features such as encryption, java/flash for virtual keyboards and stuff), and that it supports whatever security application your bank offers.
You must indeed use whatever security app the bank offers, but be extremely careful! Phishers use software that LOOKS like your bank's to steal account info from you. Always check the software publisher, the download origin and even ask your bank if it really offers software before getting it.