I think you have your timeline all messed up.

The Sword Hunt was started by Toyotomi Hideyoshi, the Kanpaku, at the end of the Azuchi-Momoyama period (also part of the Sengoku Era, around 1600). The emperor had little power at this stage and was mostly a symbolic figure.

The Boshin War occurred at the end of the Edo period (~1850) when the Tokugawa Shogunate, who had ousted the Toyotomi clan, faced rebellion from other clans that wanted to restore the Meiji Emperor to power and overthrow the Shogunate. The main causes for the rebellion included everything from anti-Western sentiment to old clan grudges.

This act is regarded as one of the major reasons for the boshin wars

As another comment addressed, the Boshin War occurred before the banning of swords which occurred in 1876.

This was of course a big deal for the Samurai class as it essentially brought about the end of the Samurai class. The samurai's swords during this time served as their badge of duty. It was basically the final nail in the head of a process that removed the Samurai class from power.

As for the Daimyo, they had already been removed from power 5 years earlier with the abolition of the han system. Whether or not the Daimyo could carry swords after this (until 1876), I'm not aware.