Summer squash including zucchini (aka courgettes) is a low-acid food. To can summer squash or zucchini, you need to do it as an acidified condiment such as pickles, relish, or crudités, and water bath or steam can the jars.
Otherwise, because summer squashes including zucchini are low-acid vegetables, they would have to be pressure-canned, and the catch is, you can’t currently do that.
What is the challenge
The problem occurs because of how mooshed up and squished together summer squash and zucchini get: the extreme high temperatures in pressure canning cause it to disintegrate and compact together, which creates density issues in the jar, which makes heat penetration unpredictable. This means that the heat required to sterilize the food might not reach everywhere adequately inside the jar.
The Clemson Cooperative Extension explains,
Slices or cubes of cooked summer squash will get quite soft and pack tightly into the jars. The amount of squash filled into a jar will affect the heating pattern in that jar and may result in inadequate processing and an unsafe product.” Preserving Summer Squash. Clemson Cooperative Extension. Accessed March 2015 at https://www.clemson.edu/extension/food_nutrition/canning/tips/26preserving_summer_squash.html
Exceptions to the rule
That being said, summer squash and zucchini are safely incorporated as an ingredient into some pressure canning recipes that have been specifically tested for how they behave during the canning process.
For instance, the USDA 2015 Guide has a recipe for pressure canned Tomato and Zucchini.  Tomatoes with Okra or Zucchini. United States Department of Agriculture (USDA). 2015. Complete guide to home canning. Agriculture information bulletin No. 539. Available at: https://nchfp.uga.edu/publications/publications_usda.html (Accessed Feb 2017). Page 3-12
Ball has a recipe for pressure canned mixed vegetables which calls for zucchini.  Ball Blue Book. Muncie, Indiana: Healthmark LLC / Jarden Home Brands. Edition 37. 2014. Page 113.
These two recipes are fine. The issue is not that summer squash and zucchini are inherently evil; it’s the unpredictable density when you process it all together by itself as a solo pack in a single jar, for which there are currently no recommendations that will cover all variables (and mistakes) to keep everyone safe. In the mixture recipes, the presence of other solid vegetables that keep their shape guards against one solid moosh lump forming in the jar which would prevent even heat penetration from occurring. Again, the high temperatures achieved in pressure canning are of no use if their sterilizing power can’t penetrate everywhere in the jar.
History of home-canning summer squash
It’s true that there used to be USDA-provided instructions for pressure-canning summer squash, but they were withdrawn in 1988  “Summer squash canning was removed from USDA complete guide [ in 1988 ]” — Andress, Elizabeth. “History, Science and Current Practice in Home Food Preservation.” Webinar. 27 February 2013. Accessed January 2015. . It’s not that it was proven unsafe: it’s that when they retested the old directions in the 1980s during a complete revision of the USDA guides, the results were not coming up consistently safe. And, when they attempted to locate the paper records holding the data behind the previously recommended processing times, the documentation could not be found:
Documentation for the previous processing times cannot be found, and available reports do not support the old process. Attempts to reproduce the old process did not result in adequate heating to ensure safety.”  Preserving Summer Squash. Clemson Cooperative Extension. Accessed March 2015. .
If you’re interested, you can read more about that USDA review period.
They also had to triage: they didn’t have a zillion dollars in funding. So, given that it was proving problematic, and that not everyone was crazy about the resulting quality anyway, it didn’t make the cut. And, owing to even less funding since then, there have not been the resources to re-tackle the topic, and consequently, there are currently no tested procedures that they feel confident in recommending to consumers as being 100% safe.
Due to the lack of tested recipes, the canning of summer squash without the addition of vinegar (for pickling) is no longer recommended.” Schmutz, Pam H. Preserving Summer Squash. Clemson Cooperative Extension. August 2007. Accessed March 2015.
So, since they can’t guarantee at what processing point it will become safe for everyone under all possible circumstances, they have withdrawn their instructions. The USDA’s modern home canning division just does not guess and say, “Give it a whirl; you might get lucky — or not.”
The Ball Blue Book also used to have directions for home canning summer squash on its own. Their recommendations were a mix of pressure canning or water bathing up to their 1947 edition (Edition X.) In 1953 (Edition 26), they switched to pressure canning only recommendations. Their last recommendations for canning summer squash appear to have been in the Blue Book reprint of 1999. (Page 65)
After that, Ball too dropped any support for home canning summer squash. If you are still using a Blue Book from that era, it’s past time to update to the current edition, as the advice in the older one is severely out of date.
So to be clear, unless you pickle summer squash in some way, there is currently no research-based proven safe way that anyone reputable is supporting to home can summer squash or zucchini (as plain products by themselves). Instead, freeze or dehydrate it (or, use it in one of the mix canning recipes mentioned above under “Exceptions”)
Zucchini. Penn State Extension. 14 July 2012.
Why is canning summer squash or zucchini not recommended? In: Frequently Asked Questions. National Center for Home Food Preservation.