The question is essentially a buying recommendation in disguise. "Is my computer good enough to record?" is pretty much the same as "What computer do I need to buy in order to record?" Additionally, this question will only be good for ~6 months until new hardware is released. While the min specs might not change much for Minecraft, if we were to generalize the question to new games, it'd change basically every month (especially since PC gamers are expecting some big leaps with regards to graphics in the near future, given the upcoming console releases).
Also, there are a whole host of things to consider beyond the software and hardware of the computer: What resolution do you want to record at? What frame rate? What are you doing with the video after you record it? Are you planning on streaming while recording? How much do you care about the video quality of the final product? (which is highly subjective, BTW.) Without this information (and possibly more), the question isn't very constructive. With them, it can quickly become too localized.
Finally, even with the specs listed (which wasn't much), and all the above questions answered, the best anyone would be able to do is say "probably". In the end, you're not going to know how good your recordings are going to be until you record something.