Well, yes, it is possible to have eye infection if you have got eczema. And generally speaking, if you get eczema around your eyes, red, itchy and inflamed patches will occur. And also, swell eyes may be caused. And according to some experts, when eczema occurs on the eyelids or the area around the eyes, rubbing and scratching should be avoided, for it can lead to potentially serious infections, such as eye infection. To treat it, you can use cool compresses to your eye area to relieve discomfort and itching, and also it can reduce swelling. Also, some eye drops are also essential.