Hi everyone, sorry to drag up an old thread, but I came across it via Google and thought I might have something helpful to add, and then people will have something thorough to read when they look up Jackie Bichons.
I have an 11 year old Jackie Bichon named Barney. When we got him from the RSPCA, we were just told he was a Bichon cross, but when my mother-in-law came across a dog that looked just like him and asked about him, we found out his dad was probably a Jack Russell. This would make sense with his personality and some parts of his appearance (I'll go into both in more detail).
He looks much more like a Bichon Frise (I've attached some photos), with very curly hair, only instead of being pure white, he has light, ginger patches on his back. He is smaller than a Bichon Frise, and has a Jack Russell's ears and face, although it can be hard to see under all his curls. Obviously though, you can get Jackie Bichons that are much more like a Jack Russell.
Before I go into his personality, I should add that we have only had him since he was 10. Before that, from what I can gather, he lived with an old lady his whole life. Having his owner die and being put in kennels was very distressing for him, so some of what I write may be down to bad experiences rather than his breed(s).
Barney is definitely more like a Jack Russell in temperament. He is very much a 'big dog in a small dog's body'. He has a tendency to bark and snap at pretty much any dog that is bigger than him, particularly when with a friend of his which is also a Jack Russell cross, so I keep him on a lead when in the busier parts of the park. They have a tendency to gang up together when they see a dog they don't know, although I've never known either of them to bite. They tend to make a lot of noise, but if the other dog responds, they run away terrified.
Saying this, once Barney gets to know dogs, he gets along well with them. Despite being 11, he still enjoys playing sometimes, particularly with puppies, and he has a toy box at home that he will get something out of when he's in the mood. You can't just get a ball out an expect him to run for it he doesn't want to, although I guess if he was younger that might be different. He also loves his walks, and you'd never be able to guess how old he is with the way he runs around. He is very fit and healthy for a dog of his age and people often mistake him for a puppy.
He can be quite a dominant dog, and has a bit of a tendency to, er...climb on top of other male dogs. We have to be careful to keep in check in home so he doesn't start thinking he's in charge. He can also be quite possessive of things like food and toys. If a dog in the park drops his ball, then Barney will take it, simply because the other dog wants it.
He's a bit of a barker at home, especially when he hears noise outside or a knock on the door. He also goes into terrier mode when the post comes and tries to 'kill' it.
He's an extremely intelligent dog with an excellent memory (for example he knows where the food will be in my parents' house even though we don't visit often, and if he finds a stick in the park that he likes, he will hide it and come back to it the next day). He sometimes uses his intelligence for naughtiness, but it has also meant that he picks up on commands easily.
On the Bichon Frise side, he's very affectionate and loves company and attention. He's never happier than when he's curled up on your lap or next to you on the sofa. He loves to be groomed and to lie on his back and have his belly tickled. He is an immediate friend of anyone who gives him affectionate, and patient with children, though you have to watch to make sure they aren't frightening or overwhelming him, because then they get a warning snap from him which can be a bit frightening for them. He also does the Bichon Frise 'buzz' thing, where he will get a sudden huge amount of energy and run round madly
The biggest problem we have with Barney is separation anxiety. He hates to be left alone in the house and will bark constantly. He's better than he was when we got him (he used to whine if we left the room) but I don't think we'll ever have him properly sorted, due to his age. However, I think this is probably down to the bad experiences he's had, and being never left by his old owner. If perhaps he'd been trained more as a puppy he would be a lot easier than he is. Saying that though, I love him to bits
I should add though, as has already been stated, with mixed breeds you don't know how much of each breed you are going to get in the dog, so you should look up information on both breeds. By going through the RSPCA (in the UK) we had a lot of info about Barney before we got him from fosterer's etc, so we knew what we were getting. If there is an animal charity near you that can give you information like this before you get the dog, then I strongly suggest you go through them, plus it's always good to give an unwanted dog a loving home.