How many minimum days will he take to complete N units of work?

Pyth, 8 bytes

tfg/*TT4

How it works:

tfg/*TT4   Q is implicitly assigned to the input.
 f         test for T=1,2,3,... returning the first successful case
   /*TT4   whether T * T / 4
  g     Q  is greater than or equal to the input (second argument implied)
t          and subtract 1 from the first successful case

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In pseudo-code:

for(int T=1;;T++)
    if(T*T/4 >= Q)
        return T-1;

bonus, 22 bytes

"should return 7 for -1"

+tfg/*TT4?>Q0Q-2Q1*4g1

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Jelly, 5 bytes

×4’½Ḟ

This uses a closed form of @LeakyNun's approach.

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Due to a lucky coincidence, Ḟ is overloaded as floor/real for real/complex numbers. This is one of the only three overloaded atoms in Jelly.

How it works

×4’½Ḟ  Main link. Argument: n (integer)

×4     Compute 4n.
  ’    Decrement; yield 4n - 1.
   ½   Square root; yield sqrt(4n - 1).
       If n < 2, this produces an imaginary number.
    Ḟ  If sqrt(4n - 1) is real, round it down to the nearest integer.
       If sqrt(4n - 1) is complex, compute its real part (0).

JavaScript (ES2016), 24 bytes

Shortened version of the ES6 variant below thanks to @Florent and the Exponentiation Operator (currently only in Firefox nightly builds or transpilers).

n=>(n-1)**.5+(n+1)**.5|0

JavaScript (ES6), 30 bytes

n=>(s=Math.sqrt)(n-1)+s(n+1)|0

Based upon this sequence.

f=n=>(s=Math.sqrt)(n-1)+s(n+1)|0

units.oninput = () => output.value = f(+units.value||0);

<label>Units: <input id="units" type="number" value="0" /></label>
<label>Days: <input id="output" type="number" value="0" disabled /></label>