Do planets strengthened by squares move houses?
For example, if I have an Taurus Sun in the 1st House, and I do a planetary spiritual square for the Sun to strengthen it and I did it when the Sun was in Leo, does the Sun can also be treated as Sun in the 4th house in Leo, or does it stay in the original house it was in prior?
Not exactly, but you will gain characteristics from Sun in Leo, as seen below:
https://templeofzeus.org/SUN.php
Leo
Proud, leadership ability, stubborn, inner strength, arrogant, dramatic, loving, hates to fail, loves to brag, loves to be center of attention, show off, generous, creative, fun loving. Flattery will get you everywhere with this sign. Step on their pride and that is unforgivable. Leos need to have fun and recreation. They are natural born actors and have a dramatic flair. Leos have strong wills and their sense of pride can make them fiercely competitive. When their pride is at stake, they will fight to the death. They must be able to express themselves freely for their own emotional and psychological health and well-being. Leos are natural born leaders.
Your natal chart is a permanent snapshot embedded in your soul. Nothing ever moves.
It is worth noting the Soul will differ from one's natal chart as you complete Runic/Sanskrit Workings and Planetary Squares, as mentioned here:
https://ancient-forums.com/viewtopic.php?f=3&t=65475&p=304146&#p304146
Even if you saw his natal chart, with x,y, and z placements, none of this would matter at his current stage of life, due to all the growth and modifications of his original birth soul.
The same applies to us. What do you think happens when you do a large working to add, for example, increased Mercury/Mannaz/Ehwaz into your soul? Your current soul would then look different from the original natal configuration.