So firstly the Flexbuilder IDE is based on the Eclipse IDE... you can download the actual Flex Builder IDE or install the Flex Plug in for Eclipse. They are both pretty much the same except with eclipse you can use it for a load of other types of development projects (namely java).
So really what your asking is 'Can I maintain a Web Application developed in Flash using the Eclipse IDE?'
The answer to this is yes and no. Of course you can use eclipse to maintain the code and the class libraries. I suggest you look at the FDT plug in for Eclipse (AS2 and 3 support). Alternatively, as long as the project you are trying to maintain, is Actionscript 3 you can still use the flex builder plug in and create a pure Actionscript 3.
The reason i say yes and no is that if you have a flash project then chances are (but not guaranteed) is that you will have FLA files that will contain animations, assets and maybe even code. If this is the case then you can not use Eclipse to manage these files.
Basically... if there are classes then I'd use eclipse and some form of plug in to maintain those, and fla's, unfortunately it's the Flash IDE for you.