Are there any products I should avoid while I have eczema patches on my hands?

Avoid allergens. Keeping your skin moist with bland lotions or ointments is essential. Common avoidable topical allergens include fragrances (organic or not), cocamidopropyl betaine (often in wipes and hand sanitizers), kathon, propylene glycol, and disperse blue. Petrolatum jelly is probably the least allergenic and most effective moisturizer for hand eczema.
Yes. Avoid bacitracin, neosporin, triple antibiotic. Avoid detergents, solvents, paint thinner, ammonia, alcohol, nail polish remover. There are lots of them.