Is that enough ?

The nearest perfect square to 1871 is $\left \lfloor \sqrt{1871} \right \rceil^2 = 43^2 = 1849$ . If Mr, John was 43 years old in 1849, he would have died at the age of $1871-1849+43 = 65 < 100$ . The perfect square before $43^2$ is $42^2 = 1764$ , he would have lived more than 100 years. The perfect square after is $44^2 = 1936$ , he would have born after he had died. Therefore, he was 43 in 1849 and was born in the year $1849-43=\boxed{1806}$ .

Notation: $\lfloor \cdot \rceil$ denotes the nearest integer function.

Let us calculate the squares of the numbers:

$40^2 = 1600 \\ 41^2 = 1681 \\ 42^2 = 1764 \\ 43^2 = 1849 \\ 44^2 = 1936$

Now ; As Mr. John died in year 1871,

so he was either 42 years old in 1764 and died in 1871 or 43 years old in 1849 and died in 1871 ;

The option of 42 years appears to be invalid ,

Hence he was 43 years old in 1849.

Therefore birth year = 1849 - 43 = $\large \boxed{1806}$

Thanks @Calvin Lin sir.

We know that:- $42^2=1764$ $43^2=1849$ $44^2=1936$ As he couldn't have lived for more than $100$ years;we can conclude that he was $43$ years old in $1849$

So his birth year would be $1849-43=1806$ So his birth year was $1806$