This sounds like spam to me. If they are tweeting about how happy they are with your competitor's product, they have no reason to switch, especially if it is a paid product. Unless the tweets indicate that they are unhappy with your competitor's product and are looking for an alternative, I don't see this gaining you any customers. In fact, it will likely have the opposite effect. So no, I wouldn't do this. And if Twitter gets enough complaints about your account, they may disable it.
You can, however, use the tweets to gain valuable information. You can learn information like what features they like the most, what features are missing that they would like to see, and how they are using the product. You can then incorporate what you've learned into your own product to make it more appealing to people in the market for a product like yours.
Additionally, you can use this information to write blog posts on your product's website to attract people that are searching for that solution in Google.
An alternate technique I've heard of people using is to bid on [competitor name] keyword in AdWords. I've never done this myself, but I know it's done. This may be a sleezy thing to do, but at least it's not spammy.
Edit: See this post that shows how terribly wrong contacting your competitor's customers via Twitter can go.