This isn't a topic that I'm honestly all that familiar with (neither the lab in a box nor HGH specifically), but I can offer some speculation.
For HGH, you're lucky in that it's actually a protein, since producing proteins with a specific structure is relatively straightforward. To grow your own, you'll need the gene in a plasmid or some other vector. For a eukaryotic host (CHO or yeast typically), you can clip the gene out of any human cell and put it in (this is far more difficult than I'm making it sound). For a prokaryotic (bacterial) host, you'll need to extract RNA from pituitary gland cells, make cDNA, and then insert that into the vector. Prokaryotic hosts are easier to work with, but do present problems with the immunogenicity of the end product; eukaryotic hosts are generally less stable over several generations and pickier about their growth conditions, but do generally provide the post-translational modifications that will keep the stuff from making you sick. Chances are that someone has already constructed a plasmid with the gene in it; which can save you a lot of effort. Get the plasmid with the gene, pop it into a suitable host, and start growing it. Purifying it afterwards can be simple or complicated, and unfortunately that's not my area of expertise. Growing the organism could be done in shake flasks, which would put it well within the capabilities of a DIYer, but purifying is more complicated. Your best bet for purifying is to use a protein A column loaded with anti-HGH antibodies, and then doing an acid elution to pull the product off the column. It's not cheap, and it's easy to mess up, but it can be done. Again though, purification isn't my area of expertise.
Now, concerning the lab in a box, the idea does sound rather appealing. What you have to realize is that synthesizing a DNA construct, getting it into a microbe, and then getting the microbe to grow while maintaining that construct are 3 different processes. Right now, for most of the oligos, primers, and other DNA constructs we need, we outsource the synthesis. Short ones (20bp or so) come back quickly, but a custom 170bp sequence that we designed took around 3 weeks to be delivered. HGH has 191 amino acids, which means that the DNA sequence is 573 base pairs long; minimum. The longer the sequence, the more likely the chances of introducing errors, and the longer it will take to synthesize in usable quantities. If there's not too many introns in the gene, PCRing it out would be the approach that I'd take, but that would add unnecessary complexity to the lab in a box concept. Anyway, before this turns into too long of a rant, the problem that you're still going to run into is in purifying the resulting protein. Some proteins are easier to purify than others, and there are some ways to make it easier (his6 tags for instance), but I'd still be very hesitant to use anything produced this way as a therapeutic. Still, a system that could pop out a microbe with a specific transgene would be very useful. For simple genes, yeah, I'd think it would be fairly plausible. If you want to synthesize something complex though such as chaperonins, or molecules that require extensive post translational modifications, I can't imagine anyone pulling it off without a firm understanding of the underlying molecular biology. Hopefully I've answered your questions.