How can I make sure that FirstOrDefault<KeyValuePair> has returned a value

FirstOrDefault doesn't return null, it returns default(T).
You should check for:

var defaultDay = default(KeyValuePair<int, string>);
bool b = day.Equals(defaultDay);

From MSDN - Enumerable.FirstOrDefault<TSource>:

default(TSource) if source is empty; otherwise, the first element in source.

Notes:

• If your code is generic it is better to use EqualityComparer<T>.Default.Equals(day, defaultDay), becuase .Equals may be overridden or day could be a null.
• In C# 7.1 you will be able to use KeyValuePair<int, string> defaultDay = default;, see Target-typed "default" literal.
• See also: Reference Source - FirstOrDefault

This is the most clear and concise way in my opinion:

var matchedDays = days.Where(x => sampleText.Contains(x.Value));
if (!matchedDays.Any())
{
    // Nothing matched
}
else
{
    // Get the first match
    var day = matchedDays.First();
}

This completely gets around using weird default value stuff for structs.