It may work, but it's a risky proposition. Headphone outputs generally have a low impedance (0.5 - 50 Ohms) and relatively high level signal (5-20 dBu). Apogee One nomimal specifications state a 14dBu output without indication of impedance.
Generally an effects return is a "line level" input, it expects (depending on the specific equipment) anywhere from -10 dBv ("consumer line level") to +4dBU ("professional line level"). So rather less "potent" signal than the Apogee's headphone output.
So depending on the specific characteristics of your amp you may be able to drive it by means of the effects return, provided you keep rather low the output volume from the Apogee. If you turn it too high you almost for sure are going to saturate the amp, and damage may even be possible.
Also, due to impedance mismatching you may also loose signal quality, namely (I think) high frequencies content. Headphones typically have an input impedance of a few dozens to a few hundred ohms, so the output of the Apogee (not specified) may be optimized for that. But that's less probably a problem, as line level inputs have typically an input impedance of around 10 kΩ or more and any source with a LOWER output impedance will probably be fine.
In summary, for the price of jack adapter (~1$/€ in eBay) I say it's worth a try, but being EXTREMELY carefull controlling the output volume of the Apogee (starting from 0, increase VERY slowly and see what happens).
Also perhaps worth trying is using the lineout from the PC's motherboard. I know the audio quality may be not as good as the Apogee's, but the signal level is probably better adapted to feed the amp. Even if you only have a single "line/headphones" output in the PC, typically these output drivers are built to compromise between several types of load (headphones, recorder inputs, etc.) and
respond in a more flexible way to the load applied to them than an optimized headphones output.
For that, you would need to route audio in the PC to the motherboard audio output. This brings in another host of possible difficulties, but apparently from the Apogee description, the bundled software would facilitate that, as well as avoid internal latency.