GZIPOutputStream already comes with a built-in buffer. So, there is no need to put a BufferedOutputStream right next to it in the chain. gojomo’s excellent answer already provides some guidance on where to place the buffer.
The default buffer size for GZIPOutputStream is only 512 bytes, so you will want to increase it to 8K or even 64K via the constructor parameter. The default buffer size for BufferedOutputStream is 8K, which is why you can measure an advantage when combining the default GZIPOutputStream and BufferedOutputStream. That advantage can also be achieved by properly sizing the GZIPOutputStream’s built-in buffer.
So, to answer your question: “Should I use BufferedOutputStream at all?” → No, in your case, you should not use it, but instead set the GZIPOutputStream’s buffer to at least 8K.