You don't need to encrypt the DNA sequence in order for other people to be unable to use it.
A DNA molecule has no intrinsic meaning, meaning that while you might be able to discern patterns, you will never get all of the information of an organism purely by looking at its DNA.
Let me explain this with an analogy to language. If I gave you a book in a language you don't read, you would be incapable of understanding it. You know it's a book, and you know that it has meaning, but you can't reverse engineer it without testing things. For example taking a sentence and repeating it to someone who does speak that language, or re-arranging words and testing if they have the desired effect when you speak them to someone who understands them.
A code, in this case the genetic code, requires a decoding system in order to contain meaningful information. In biology, the decoder is constituted of many other parts of the cell, including but not limited to the transcriptional machinery which transcribes the DNA to RNA, the translational machinery which translates the RNA to proteins, the epigenetic machinery which controls which regions should be transcribed, cell signalling pathways which modulate the previous systems based on external and internal factors, and so on.
So it all comes back to the egg cell. It's not just a DNA molecule on its own. It's a DNA molecule in a very specific cellular context, and in the case of metazoans it's also a cell in a very specific organ context. A human egg cell can't develop in your bladder. 29
It has certain requirements which are (typically) met by the uterus.
If your creatures have sexual reproduction, you can engineer a breeder strain of the creature which gives birth to sterile progeny. To avoid the possibility of in-vitro fertilisation you can engineer them to give birth to a progeny lacking the equivalent of a uterus, or you can systematically perform hysterectomies before selling them.
If you really want it to be incredibly opaque, then you have to modify the way the various machineries I mentioned earlier work:
* Change the start and stop codons by modifying the transcriptional machinery to recognize different patterns. You'll also need to alter the transcription start site sequence.
* Scramble the genetic code by making the various tRNAs correspond to different amino acids, which will force would-be thieves to conduct experiments in order to figure out what your proprietary version of the genetic code corresponds to.
* If you're using pre-existing transcription factors, perform directed mutagenesis experiments to make them recognize different sequence motifs.
As for making the DNA difficult to read and interpret, here are a few ideas:
* You should include many very long stretches of repeated regions in its DNA, as well as transposons. These make it much more difficult to assemble an organism's full genome. We're starting to be capable of tackling that problem with the third generation of sequencing methods which generates long reads that can resolve these repetitive sequences.
* Add many pseudogenes to confuse interpretation, these are DNA sequences that look like genes and are recognized by automated tools but aren't functional as such.
* Systematically scramble the genomes, in some single-celled organisms it has been shown that you can cut and randomly ligate large stretches of the genome without any major effects on fitness. If you don't mind each creature being uniquely sterile it might be possible to start with a freshly re-arranged egg cell without too many negative consequences for the creature's health. You probably do want to keep specific syntenic blocks together but if you define the acceptable regions to cut then it should be fine.
Ultimately you want to make the cell different enough to ensure that in order to steal your creature, one would have to study it for so long that it's easier to just design their own original creature. Don't forget to make sure the creatures released to the public are incapable of reproducing. 1008