Hi there, hope you are enjoying BYC!
I kept goat for while... my chickens also LOVED scratching through the goat shed. Some even started laying there.
The biggest problem I ever faced was the goaties relentless attempts to get to the chicken feed... they just LOVE grain. But it could kill them ultimately if they ate too much. Are they whethers or does (sp?) or breeding?
There are certain parasites that MAY be passed between them, but most are host specific. Getting fecal floats and treating with the correct med for the species you have
IF needed, and general good husbandry can go a long way towards prevention of most stuff. Doing "group" fecal floats (several samples in one test) a few times throughout the year can help you to know YOUR internal parasite load as it is effected by so many things including weather, stock density, wild life load, and even soil type. Most species will never be seen in droppings as only large round worms and tape worms may ever be passed in droppings. All other will stay int he intestines and only their microscopic oocysts (eggs) will pass in droppings.
Sounds like your "young roo" to hen population might be a bit off kilter and I would consider re-homing or eating some if not all of the boys.

1 is more than enough for how many ladies ya got. Young roosters and hens since you mention terminology, are known as cockerel or pullet until they reach 1 year old. It's most important when discussing health and behavioral issues. But it's also great to be on a learning adventure!
Congrat's on your
blessed retirement!
Adding your general location to your profile can help peeps make the best suggestions possible at a glance int he future.
Please... if a specific comment you saw makes you wonder... post it so we can confirm or deny your cause for concern.