Yes you can, but you are going to have to maintain the salinity levels and change the water periodically. I don't hatch brine shrimp anymore myself, I just buy the adult shrimp as needed. Even so, if the shrimp are around for more than 2 weeks, I change their water as it will foul.
To maintain the salinity level with the water you already have, make a mark on the bottle where the water level is. Then all you have to do is add dechlorinated tap water back up to the line.
I change the water as it isn't a problem for me to get saltwater as I have a salt tank.......but unless you have a salt tank this could be a problem for you. If you decide to mix up your own water you will also need a hydrometer so you can get the salinity right. What might work out better for you is to get a small sponge filter and attach it to the end of the tubing. That way at least, it should collect some of the detrius and you can clean it once a week or so.....
Then again, I'm not familiar with the brine shrimp kit.....did it come with several pre-packaged salt mixes?