Can an inguinal fungal infection be transferred to the eyes inadvertently?

Not usually: The lids are not moist enough. The eyeball itself is too well protected. Contact lens wearers may be liable for fungal corneal ulcers. These fungal infections need moisture, low oxygen and low light. I assume you are not immune deficient.

Spreading fungal inf: Some dermatophytes (skin fungal organisms) can be spread on the hands, cosmetics/body creams or on clothing/ towels from one area on the body to other areas. Towels need to be washed in hot water & not shared (especially in health clubs/gyms). Good hand washing is a must. Other concern would be a contact dermatitis around the eyes from the antifungal agent used to treat the inguinal infection.