YES!!! Apparently the 'stay logged in for 2 weeks' is just another useless appendage on the login page. 
Apparently if you are not in constant activity online (i.e.. MyVerizon) 
then your connection is killed in lieu of some such security issue Verizon feels will leave you vulnerable even though You already knew the risk
...At least one would hope you'd know.
FYI leaving any connection wide open online results in leaving a vulnerability hole to 'your' whole network ...biet one computer or many ...where by some savvy web users AKA hackers might gain access. And all the while I thought I actually logged into a secured network.
Anyhow, so while just simply closing access to that port might suffice they seem to just disconnect you from their network instead, resulting in this re-login effect. 
This phenomena seems to happen between access to other Verizon services as well (i.e.. MyAccount). I have to login for access to e-mails and then again to gain access to 'My Account" and then again if I wish to continue back at "MyVerizon". It's nuts!!! But I'm kinda just going with the flow ...like I have a choice.
I believe if their was a way to trigger activity via your router, the end result might keep you connected. I don't know much about these Verizon routers however AOL routers used to get triggered periodically keeping us connected via software. I would think since our network access point is now actually in our home, there ought be a way to tell Verizon that we are still logged in. I do know for a fact that VZ router will log you out after a period time due to inactivity, resulting in the very same re-logon effect.
I mean I did knowingly open up this specific connection with good reason, ought I also be able to decide actually when this connection be closed as well. This is the "hows & whys" I derived the aforementioned conclusion. Hence Verizon is in more control of my connections than I am.
