I have found it to be the case, but in varying degrees. My step-son is 13yrs old now, has awful allergies that we have to carry an epi-pen around for and he struggles with awful eczema to the point that he often bleeds from it, especially in the creases. My 23mo old daughter has eczema as well, but not nearly as severe. We noticed it pretty early on, especially that the severity came and went with her diet. We have had her tested and she isn't allergic to foods, but she does have some pretty bad food intolerance. She is milk and soy protien intolerant (MSPI), and she also seems to be intolerant of bananas (she seems to be growing out of this a bit that she can have one every few days or so now, and anything very acidic like citrus fruits or kiwi (the worst - horrible bleeding rashes). To make it even more odd, these intolerances vary depending on preparation meaning she can have milk if it's been cultured (yogurt & cheese), she can have soy oils, can have bananas if they are cooked, essentially anything that breaks the protiens down. If she eats something she is intolerant too, she will get red, raised scaly patches, almost like hives, starting from her legs and working their way up her torso, on her back and her arms, and sometimes on her face. I have prescription hydro-cortisone cream for when it really flares up, but most of the time the best thing has been to slather her up with Aquaphor (this is a life-saver, promise you). She also will get these horribly painful, raised diaper rashes that the skin will crack and bleed from the poop when it touches her skin. This rash ONLY appears after she eats something that she is intolerant from. Also, especially with milk, she will get very irritable (they kept saying she was colicky and that she had reflux, as soon as I started treating her diet for MSPI it all dissapeared), a very bad runny nose, and this mucusy, liquidy diarrhea. The other thing was, she was very small for her age (on the boarder of failure to thrive) but as soon as she went on to Nutramigen she started to grow substantially, and when she moved more firmly onto solids and then onto rice milk - she's more than doubled at each check-up. I'm starting to think her 2yr checkup will be higher than 50% - which is a HUGE deal for her. The other tricky thing about something like MSPI (and I'm not saying this is what is going on with your son, but perhaps it is something akin) is that if you are breast feeding and you yourself consume a milk or soy product, the protien will travel through the breastmilk and affect the baby, and it can take a week or more to fully clear out of your system.
I mention all of this because eczema is an auto-immune disease, and sometimes it's nothing more than the baby is super sensitive to stuff in soaps or lotions, but it could also be an indicator that you need to be more observant. I started DD on solids at 4 mos (again, because of her dietary issues it was better for her to be on solids) and it's amazing what this girl will eat. I was able to introduce a wide variety of foods, but I just kept a log of what I fed her, waited a few days, and wrote down anything I noticed at all - change in stool, change in skin, temperment, etc. and I was able to quickly figure out what was good and what wasn't. The other thing, I think the main thing, is also that I made everything from her from scratch (BEABA is AWESOME! - I was working full time and I still could do it because it's so easy with it) so I knew exactly what she was eating. For example, even the organic squeeze pouches can cause her an issue because even though it's just "apples" it has a touch of lemon juice to prevent spoilage. Most prepared foods, especially baby foods has some sort of milk product in it (butter, whey protien, cassien protien, powdered milk, etc) - even something as simple as "pasta in broth" will have a milk product in it. So, get in the habit of reading any label no matter what the brand, and see if the eczema worsens, even a few days after he tries it.
Anyway, I found the internet to actually be my best friend in helping to figure out what was going on because there just isn't a lot of literature on MSPI. Perhaps with some research and using a log book to track when the eczema worsens and gets better can help you figure out if it's diet (your's included if breast feeding) or something else.