Could be the router.
I used to have an old Linksys BEFSR41 wired router, & it started getting flakey after about a year. Random connection losses (often in the middle of downloading something or using XBox Live

). Also, like your BEFW11S4 problem, the lights would all go red briefly.
(seemingly) Solved it for a while by turning it off for a good 5mins or more & then doing a factory reset. Didn't do it for ages after that. When it did start to do it again, a firmware upgrade helped.
Is the router old? Maybe that's a factor too. Is the firmware up to date?