With an unknown rooster it is hard to be sure. That could be a leghorn which means he would add white eggshell genetics. If it is a leghorn the odds are pretty high the eggs from the leghorn/RIR would still be brown though the shade might be lighter than the RIR’s are laying.
Eggshell color genetics can be pretty complicated. There is one gene pair that establishes whether the base color is blue or white. The odds are huge that that rooster will add a base white gene, just like the RIR, so that part is pretty simple. To simplify, brown is just brown on top of a white egg like green is just brown on top of blue.
But there are a lot of different gene pairs that affect the shade of brown that might be added. Last I saw there were 13 different brown shell color genes that had been found but who knows how many are yet to be discovered. Some are dominant, some recessive, and some only act when some other gene is present or absent. That’s why you can get such a variety of brown shades. There is even a gene that bleaches out a certain brown gene so if both those genes are present they cancel each other out. There are a tremendous number of possible combinations. That’s why you get so many different shades of brown eggs. It is also practically certain that the different RIR hens have different combinations of the genes that effect brown.
What I would expect from that cross is brown eggs, probably lighter in shade than the RIR’s are now laying. You will probably get a range of different shades of brown. But don’t be shocked if you get an egg that is actually a bit darker instead of lighter. It’s even possible you could get an egg that is basically white but this is pretty unlikely. Still, some could be pretty light. You are just going to have to hatch some and see what you get.
You did not ask but you might get some interesting colors or patterns from the chicks. There are two different ways genetically to get a solid white chicken like that Leghorn. One way is Dominant White on Black. There are different ways to make black genetically so you can get some surprises here too but if that rooster is dominant white on black the odds are pretty good you will get yellow chicks that make white chickens.
The other way to make a solid white chicken is Recessive White on top of anything when the recessive white genes pair up. Recessive White is really strong when it pairs up but really weak when it doesn’t. Since you don’t know what might be under recessive white you could get a real rainbow of different colors and patterns from that cross. Again the only way to find out is to hatch some chicks.
Good luck if you decide to hatch some chicks. It could be kind of boring as to what the chicks look like and what color eggs the pullets lay or it could be really interesting.
