The "active" flag is a physical bit-switch set in the MBR partition table which indicates to the BIOS where the boot files reside so that the boot can chain via those to the OS(s). Changing the flag will point the BIOS to the wrong place with inevitable consequences
I used the 'skip the boot menu' option in easybcd and connected up the old hd into the new tower. When I boot into Win 10 I can access all the drives, but when I try to boot into Win 7...
If you skip the boot menu, how are you getting a choice to boot into W7 ?
Windows when you (or the OEM) install it will tailor itself to the hardware it detects and build a specific set of hardware drivers for the mobo chipsets and other attached cards found.
If you take that OS and place it in a new hardware environment, it's highly unlikely that it will boot successfully without what XP used to call a "Repair Install". In Vista onwards that option disappeared and you had to do an "In-place upgrade" or somesuch description. All those options require the original installation media which will efectively reinstall the OS whilst retaining your personal files and programs.
If the old OS came pre-installed on a PC, it will have an OEM EULA tying it to the hardware it was supplied with. (i.e. the old PC owns the OS, not you)
Transferring it to new hardware (unlikely to work anyway for the reason given above), even if serendipitously successful would result in activation being denied by MS because Windows would detect the mobo serial number change and know that it doesn't have a valid licence to work on that h/w.
If you have the W7 installation DVD, you should boot into W10, add an entry for W7 into W10's BCD then reboot and choose that entry. When it fails to boot and asks for the W7 DVD to fix the problem, follow its instructions.
If the W7 EULA is OEM however and you don't have a W7 DVD, you won't be able to use your old OS on your new PC.