Using the condition $a + b + c = 1$, we obtain
$$a^2 - 2bc + 2c = a^2 + 2c(1 - b) = a^2 + 2c(a + c) = (c + a)^2 + c^2.$$
Similarly, the left-hand side of the inequality becomes
$$\sqrt{(c + a)^2 + c^2} + \sqrt{(a + b)^2 + a^2} + \sqrt{(b + c)^2 + b^2}.$$
By the triangle inequality, this is bounded below by
$$\sqrt{((c + a) + (a + b) + (b + c))^2 + (c + a + b)^2} = \sqrt{5}.$$
Equality holds when $a = b = c = \frac{1}{3}$.