I have red, swollen, itchy eyes. Could this be due to makeup allergy?

Yes. Red, itchy, swollen eyes may indicate an allergy to almost anything...Laundry detergent, dryer sheets, eye makeup, lotions, mail polish and may other things.
Absolutely. But there are other causes of swollen and itching eyes: seasonal allergies, pet dander, and even some forms of pink eye. Antihistamine eye drops can decrease the allergic response, but it is best to have an eye doctor check you out before starting your own medication regimen.
Homeopathy. Consider homeopathic treatment for allergic symptoms. There are a few homeopathic remedies with great affinity for eye symptoms associated with seasonal allergies.