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Leaves drying one week after planting is not a good sign. Handling and planting technique are important, but the most vital is adequate water at the time of planting, and a good water management program to follow.
If you think that the soil root ball and soil backfill in the planting hole may not have been soaked thoroughly at the time of planting, then try this. Poke a running water hose end down into the loose soil backfill around the outside of the root ball. Use a stick to poke straight down in multiple spots and try to settle the soil and fill any air pockets that might have developed in the backfill. Follow up with deep watering a couple of times a week. Give it a month to respond with new growth. If it continues to dry and die back then it may be lost.

Your bottlebrush shrubs might have a nutrient deficiency. If the younger leaves are paler/yellower than the older leaves, it's most likely an iron or manganese deficiency. With iron deficiency, you will usually see the leaf veins staying green while the rest of the leaf turns yellow. If all the leaves are turning yellow, including the leaf veins, or if the older leaves trun yellow first, it is more likely a nitrogen deficiency.
https://www.gardeningknowhow.com/plant-problems/environmental/leaf-chlorosis-and-iron.htm
http://web.extension.illinois.edu/cfiv/homeowners/070828.html

If your shrubs are not in full sun, the yellowing could also be caused by low light levels.

This can be a big issue. This article will give you tips on how to control these pesky creatures: https://www.gardeningknowhow.com/plant-problems/pests/animals/get-rid-squirrels.htm

If it has been excessively cold, then this will be the best option. At least to cut it back until you see living tissue. This can help to regenerate the bush after a hard frost.

Here is an article on their general care:https://www.gardeningknowhow.com/ornamental/shrubs/bottlebrush/growing-bottlebrush-plants.htm

Yes, just trim off any dead material, and it will recover on its own. This article will give you more information: https://www.gardeningknowhow.com/ornamental/shrubs/bottlebrush/pruning-bottlebrush-plants.htm

Can you include a photo? This sounds like a bacterial/fungal infection but I want to be sure. It is always good to add dolomitic lime and wettable sulfur to the area once per year anyways, but this is also a good spot treatment for many infections in the soil. I would try this first, as it will likely clear it up.

Again, if you can include a photo, I will be able to better guide you on what should be done from here.