Actually, the answer is NO. While those SATA to eSATA brackets are common (Asus even shipped one with my motherboard), they are out of spec. eSATA and SATA have different voltage ranges. If you put the external drive in a true eSATA housing, then you are running signals between two ports that are technically incompatible. The voltage ranges of the two specs overlap, so it will work some or most of the time, but the risk of failure is pretty high.
If you run the cable to the drive directly, then you are running SATA to SATA, and the only spec that you are violating is having the SATA cable run out of the case.
Either way, it is out-of-spec and thus carries risk.