To find out when Henry will collect his 500th egg, we can first calculate how many eggs he collects each day from his 37 chickens. Since each chicken lays 1 egg per day, Henry collects:
\[ 37 \text{ eggs per day} \]
Now, we need to determine how many full days it takes Henry to collect 500 eggs. We can do this by dividing the total number of eggs he needs by the number of eggs he collects per day:
\[ \text{Number of days} = \frac{500 \text{ eggs}}{37 \text{ eggs per day}} \approx 13.51 \]
Since he can't collect eggs for a fraction of a day, he will collect eggs for 14 days.
Now, let's calculate the total number of eggs collected in 14 days:
\[ \text{Total eggs collected in 14 days} = 37 \text{ eggs/day} \times 14 \text{ days} = 518 \text{ eggs} \]
On the 14th day, he will collect 37 eggs, bringing his total to 518. However, he only needs to collect 500 eggs.
To determine how many extra eggs Henry has at the end of the day when he reaches his 500th egg, we can calculate the number of eggs he collected before reaching 500 eggs.
On the 14th day, before he collects the 37th egg, he already has:
\[ 518 - 37 = 481 \text{ eggs} \]
Thus, on this day, he will collect the next set of eggs until he reaches 500:
\[ 500 - 481 = 19 \text{ extra eggs collected on the 14th day} \]
Finally, after collecting his 19th egg, he will have:
\[ 518 - 500 = 18 \text{ extra eggs} \]
So, the number of extra eggs Henry has left over at the end of the day when he collects his 500th egg is:
\[ \boxed{18} \]