Can a virus be modified and used to sterilize mammals?

Cool idea. Let's use your rat example. Hair pigment is produced by melanocytes in the hair bulb. Hair color is influenced by many different genes, and each gene contains different alleles (versions). On top of that, although some pigments may be proteinaceous (protein by nature), most are not proteins. Hence, it would be a very hard task to program a virus to eradicate rats based on their hair color. 

Hmm I thought so.

Okay, how about if there were a bit more physical differences. Say maybe dog breeds? We want to sterilize Chihuahuas but not Min Pins, or even sterilize Poodles but not German Shepherds?

Firstly, you're putting the carriage in front of the horse here, you have to do some serious googling on viruses before heading out to engineer one.

What exactly do you mean by "design" ?  Viruses are arguably one of the oldest organisms to evolve on earth, with billions of years of evolution on their side, designing one seems simple because they look simple from afar, but in reality they are small pieces of genetic code that pack a huge punch, evolutions brilliant multitaskers. That said, if you were to take a look at the protein folds of numerous viruses, they would look a lot less like blobs and more like glorious architectural marvels, because indeed they are.

First you have to ask yourself what is a virus? Viruses don't come in a single flavor, they come in many flavors!! Just look at the "Baltimore classification" for viruses, you'll find there are many types with very unique differences on how and where they replicate in a cell.

First off you can google Viruses in mammals + sterility. I tried to put the link but the site didn't let me.
Second, you can google rodent fur color to get an idea what hair color consists of in mammals.
Thirdly, look for a paper with Sendai virus and rats. This virus is a paramyxovirus, like that of Mumps virus, which causes testicular orchitis in human males. It's a really interesting paper.

Moving on, if we took Sendai virus for example, it causes reproductive problems in rats. Well, how do these viruses cause this kind of damage? In SV's case, it was directly injected into the testes of rats, the rats were culled at four timed intervals to see the evolution of disease pathology in the testes. Like many of these infections in the reproductive organs, it's not the pathogen that causes gross amounts of damage by the immune response they mount. The inflammation in the testes was extremely rapid and immense, overwhelmed by leukocytes, the mounted immune response is ultimately what causes the most damage, mostly by inflammation and collateral damage. If the inflammation in the testes is prolonged, the damage can become much worse, which leads to real reproductive problems. Obstructions caused by scar tissue and damage, ect. So, it's not all blood and gore that these viruses cause down there. It's our bodies that they manipulate against us.
What that means for your idea, well, look at the evidences. There would have to be a clear shot between color and reproductive organs.  (I'm making this up), Suppose you have X colored mouse whose color is associated with a novel gene that codes for extra surface proteins, those of which interact with "ABC virus" (made up name), then ABC virus coming into the body would be able to infect a higher volume of cells than it's Z colored mouse counterpart with less of the surface protein. Do you see how you would have to look at the cells you expect to target, the mode by which infertility is accomplished, and if there is a relationship between those cells and hair color. IF you don't know, right now, what the connection between the two are, then you couldn't engineer a virus to do this because you have no means of actually executing it. What would you program the virus to attack? Melanocytes of grey haired mice? What would that accomplish?

You would be looking for the difference linked to hair color in the form of a selected advantage for viral infection of one mouse color over another. That's a tall order. Good luck to you and I hope I helped. If you're curious about how people work with viruses, read lots of scientific journals, they outline all their procedures. If you want to know what type of equipment, same thing, grab a journal of molecular biology or cell or nature. Scientists who do this work are all of the above. From biochemists to virologists, it really doesn't matter what area of study you come from, so long as you have experience in the field. Any lab working to reprogram a virus to do something specific is going to use skilled hands. Graduate students, assistant professors, post docs, professional scientists with Doctorates, you name it. You have to be able to design experiments that test your hypothesis well, and science is expensive! IF you write a grant proposal with nothing more than a wild idea, you're in for a rude awakening. You must have observed something first, unless you're one of the lucky ones that finds what they're looking for before they know what 'it' is. lol.

Thanks for the response. Much appreciated.