I'm not presently using Eagle and don't have a working installation on my computer, thus I'm unable to inspect the design files.
Generally, there should be at least at meshed ground net. Power supplies should preferably have bypass capacitors at each IC supply pin and don't need to be meshed. In my view, the filter topology is simple enough to have room for power supply and ground in a two layer design. A well considered placement is required, however.
As previously mentioned, 0403 seems unresonably small for the present application. You'll possibly face difficulties to get low T.C. capcitors (NP0 or similar dielectrica) in the intended capcitance range as required by a good filter. You would rather go for a smaller quad OP package (TSOP or chip scale package) to save board space.
Two sided component placement would allow ultimate circuit density, but doesn't make much sense without changing to a four layer PCB.