Okay, this is my third attempt to answer your question, the first two failed because of my laptop crashing for weird reasons.

So to better answer your question, I am going to separate it in three parts, the first one is about the region, then about Schaffausen and finally about the enclave :

- In order to understand the whys and hows, it's important to know (or remember) that the region has been part of the Holy Roman Empire (HRE), and that the concept of Switzerland is pretty recent : the Federal Pact was signed in August 1291 but was only made with 3 cantons ( Uri, Schwytz and Nidwald, the second giving the name of the country). You'll even notice that there is a difference between the "common name" and the "formal name" (Switzerland and Helvetic Confederation). That difference is a perfect example of the particularity of the country. The first one refers to one of the canton and never really been a concept to designate the region (which was more considered as part of the cultural region and the country of Swabia). The second one refers to one of the first cultural group living in part of the region : the Helvetii.
The denomination of the country's name "Confederation" is also a part of the answer : in contrast with other countries that have been build through wars and moving borders, Switzerland (not so neutral if you look at their history) has mostly grew through the willingness of other counties, city-states, cantons to join, explaining the popping out of some cantons.

- Schaffhausen was a small city-state at first (with some land outside of the city, so more like a small county). The city became recognized as a City (in this case it could be defined as a center of trade for its region and a place that have a lot of wealth) by the HRE Emperor and joined the HRE in 1190. The city is even recognized as an "Imperial City" (ancestor of the "Free Cities") in 1330. Because of this high concentration of wealth, the bourgeoisie's population boomed and the Duke Leopold of Austria gave a Charter to the city stating that half the High Council, the Small Council and even the Court House had to be composed by a half of noble and the other half of burgher (bourgeoisie). However, in 1387 this distinction was removed.
I'm talking a lot about that because that's, in my opinion, one of the roots of why Schaffhausen became part of the Confederation.

Because of the revocation of the Charter in 1387, the bourgeoisie of the country fought against the Austrian army. In the end, they managed to get some of their rights back and even bought back others later on in 1411. By that act, Schaffhausen put a burgher at the head of their state, making it a Republic. The Republic took advantage of the imperial ban on Frederic (duke of Austria) to buy the rest of all their rights in 1415 for 30 000 ducats from Emperor Sigismond. The Emperor sworn to give the right back to Frederic when the ban will be removed, but he did not.
Frederic unhappy and claiming that Austria had authority other Schaffhausen went to war with the young Republic, however the republic already had alliances with the majority of the Swiss' confederation and managed to push the evil Austrian Duke back. In 1501, Schaffhausen officially joins the Confederation!

- The enclave of Büsingen am Hochrhein would be a way shorter answer since I didn't find much detail about it. From what I gathered (book at the end of the comment and the sources that I had are no more available), the separation is from the conflict between Schaffhausen and Austria. The bailli of Büsingen had arguments with protestant priest which result in a drama (the bailli threaten to call the Catholic Church to intervene) and he ended up being kidnapped. When he got out he simply sworn that the city will never go to Schaffhausen's authority. Apparently that "oath" has been respected. In 1918 a referendum was made in which 96% of the population agreed to join the confederation, however the confederation refused to cede advantages to the population, so the Confederation refused. Since that date, all propositions have been refused by Switzerland.

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In conclusion : Schaffhausen is not part of Germany because it was under influence of Austria for a long time (so was most of the region) and the political move created tensions and resentment, the allies of the republic weren't the other german states but the swiz ones, gathering them into the confederation. And for the enclave, it's more or less the same concept : resentment. Moral : Stubbornness can change borders, think about that next time you yell at a donkey for not doing what you want ; who knows, you might end up in the barn and the donkey in your living room.
For the language, that's probably more due to the cultural region, I hope you got the answers you were looking for :) and I hope I managed to be clear and understandable.

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My main source for my answer has been a book (free and online) named "La Suisse Pittoresque" by Charles Schaub, Eusèbe-Henri-Alban Gaullieur. It's available in French, I don't know if it is in other languages.