The testicles are not covered in "taste bud cells". The testis are within the scrotum. The anatomy of the scrotum is described below (from the 4th Edition of Urologic Surgical Pathology). Epidermis and dermis are layers of skin. Here is a link to an article that explains where the internet buzz came from. Hope this helps!

The scrotum consists of skin, dartos muscle, and external spermatic, cremasteric, and internal spermatic fasciae. The internal fascia is loosely attached to the parietal layer of the tunica vaginalis. The epidermis covers the dermis, and the deepest layer of the dermis merges with the smooth muscle bundles of the dartos tunic. Although scattered fat cells are present, there is no subcutaneous adipose tissue layer. The dermis contains hair follicles and apocrine, eccrine, and sebaceous glands.

We understand and perceive tastes with the help of protein molecules. There are two sets of these molecules, one that is present on our taste buds called receptors and another on the food that we are eating, let’s call them ligands. While eating these, the molecules form the food interact with the molecules on our tongue. Technically the ligand goes and attaches to the receptor. Different receptors have different ligands. This is the reason we can perceive different tastes so precisely, for example, vanilla ice-cream is so distinctly sweet, and noodles are so savory and salty.

We can think of these receptors and ligands molecules like a lock and key. Where the receptor is a lock, and the ligand is the key. In general, the key opens the lock because it fits in perfectly. However, in this case, one the ligand interacts with its receptor perfectly, it sends a signal to the brain and, depending upon which ligand interacted with which receptors the brain analyses the tastes.

Now to answer your question, I have mentioned above that these receptor molecules are made up of protein that resides on the surface of the cells. Various kinds of receptor proteins have been identified, TAS1R3 and GNAT3 are two of those. Scientists knew that these were only present on the tongue as a part of taste buds and hence did the work of identifying taste. However, they recently found out that in mice, these proteins are also present on the surface of their testicles.

If you make a mouse in a lab that does not have these proteins, they do not reproduce, which made them look into their testes and led to the discovery of these proteins present on their testicles as well. However, we do not yet know what they are doing there. We know they are responsible for the proper growth of the sperms for the mice to reproduce. Like on the tongue, they send signals of taste; on the testicle, they must be sending out some critical signal as well, which is not taste but related to the formation of sperms. Hence, I am not sure if we can call those cells as taste buds even though they do share similar proteins.

Also, this study was done on mice, and we do not know how far we can extrapolate it to humans. Being genetically very similar to mice, the scientist does think that might be one of the possibilities that humans also have them, but the opposite is also equally possible.

Ms. Branch, I found your odd question to be quite thought-provoking. I'd like to share one study and two news articles with you and your class. Interestingly, taste buds have been found inside a wide variety of organs. For example, the pancreas, lungs and brain. Indeed, inside the testis are taste bud cells that produce a certain protein. According to the study, this protein links to fertility in mice. Fun fact, the taste cells are able to taste umami, found in soy sauce.

Study: here.
#1 News: here.
#2 News: here.

I found this interesting article for you: here.

“In mammals, the sense of taste helps in the evaluation and consumption of nutrients, and in avoiding toxic substances and indigestible materials. Distinct cell types expressing unique receptors detect each of the five basic tastes: salty, sour, bitter, sweet and umami. The latter three tastes are detected by two distinct families of G protein-coupled receptors: T2Rs and T1Rs. Interestingly, these taste receptors have been found in tissues other than the tongue, such as the digestive system, respiratory system, brain, testis and spermatozoa."

..." Interestingly, these taste receptors have been found in tissues other than the tongue, such as the digestive system, respiratory system, brain, testis and spermatozoa. The functional implications of taste receptors distributed throughout the body are unknown."

This article cites the one mentioned above.

From one of our favorite participants in ScienceLine:
I used to give a lot of “lectures” at local elementary schools about the planets, space travel and related matters. The most common question in all of my ~ 20-30 years of doing this was:
HOW DO THE ASTRONAUTS GO TO THE BATHROOM IN SPACE?