Our dog died very suddenly (hit by a car) when my twins were about 3-1/2. A few months prior to our dog's death, the girls saw a dead squirrel in the road and my husband explained death to them at that time. When our dog died, we were honest about what happened. They took it in stride, really. They asked a lot of questions and participated as best they could in a funeral we had for him. They were a lot less upset than we thought they'd be.
I did get some books about a pet dying. I bought several on Amazon and none were all that great. Mostly they didn't apply to our situation in that his death was sudden and violent, not the typical situation of an older dog being put down where the kids get to say goodbye first.
About telling them the dog was sick, when my parent's dog died, I said "he was very old and very very sick, much sicker than you get when you get a cold." They don't seem worried that they're going to die when they get sick. My twins are 5 and they've learned over time that sometimes bad stuff happens and kids and younger people die. They've just been exposed to that through life. Death is a very real part of life and I do not agree with shielding kids from it. I think being honest, direct and providing answers as best you can with comfort is the way to deal with these kinds of things. It is hard though.