Replacing a natural diet of pasture and insects with grain-based feed increases the concentration of pro-inflammatory lipids in egg yolks.

Study: Nutritional supplementation of ahiflower seed and press cake in laying hens and its effect on production performance, nutrient digestibility, egg quality, and yolk omega-3 fatty acid enrichment during mid- to post-peak production phase

This study didn't compare grain-fed hens to free-range hens, but it did show that changing what hens eat changes the fats in their eggs—some of those changes make the eggs more likely to cause inflammation. So it supports the idea that what hens eat matters for egg health.

Study: Fatty Acid and Antioxidant Profile of Eggs from Pasture-Raised Hens Fed a Corn- and Soy-Free Diet and Supplemented with Grass-Fed Beef Suet and Liver

Eggs from hens fed grains had more unhealthy fats that cause inflammation, while eggs from hens eating grass and bugs (or similar natural foods) had healthier fats. So yes, switching to grain feed makes egg yolks more inflammatory.

Study: ω-6 and ω-3 Polyunsaturated Fatty Acids: Inflammation, Obesity and Foods of Animal Resources

When hens eat grains instead of bugs and grass, their eggs end up with more fats that cause inflammation and fewer fats that reduce it. So yes, grain-fed eggs are more inflammatory.