Simply, can you be on an IFR flight plan (VMC) day + night, solo (no PAX or instructor) to fulfill the CAX XC day/night requirements? Reading the regs it kinda looks like you have to be VFR, but maybe just VMC.
Which cross country requirement. You said "day/night." The only day/night cross country requirement in 91.129 is a "training" requirement, i.e., dual. The long solo cross country requirement doesn't specify day or night.
Neither is required to be VFR. I'm not sure what you read that leads you to think so. Maybe you could quote the regulation so we know if we are on the same page?