Is it possible for $\Delta x$ ($\sigma_x$) of any free particle wave packet to be decreasing at any time?
If $\Psi(x,t)$ solves the Schrodinger equation, so does $\Psi^*(x,-t)$ , so no, there is nothing at all that must increase.
We reformulate OP's title question (v1) as follows:
Show that for all possible wave packets $\psi(p,t)$ of a free particle, the position variance $$ {\rm Var}(\hat{x}) ~=~\langle \hat{x}^2\rangle-\langle \hat{x}\rangle^2\tag{1}$$ decreases in at least some interval $[t_1,t_2]$ of time.
As Mark Eichenlaub correctly observes in his answer and comments, a solution with decreasing ${\rm Var}(\hat{x})$ in $[t_1,t_2]$ can be mapped by time reversal symmetry to a solution with increasing ${\rm Var}(\hat{x})$ in $[-t_2,-t_1]$. Here it is used that the Hamiltonian $\hat{H}=\frac{\hat{p}^2}{2m}$ for a free particle, and the position operator $\hat{x}$ both commute with the time reversal operator $\hat{T}$. See e.g. my Phys.SE answer here for further details.
However, time reversal symmetry does logically not rule out alone the possibility that a solution has monotonically increasing ${\rm Var}(\hat{x})$ for all time. (It only implies that if this is the case, then there will also be a solution with monotonically decreasing ${\rm Var}(\hat{x})$ for all time.)
Explicit calculation. Consider an arbitrary wave packet $$ \psi(p,t)~=~A(p)e^{-i\theta(p,t)} \tag{2}$$ that represents the general solution to the time-dependent Schrödinger equation in the momentum representation. Here the angle $$ \theta(p,t)~=~\theta_0(p) + \frac{p^2 }{2m}\frac{t}{\hbar},\qquad A(p),\theta_0(p)~\in~\mathbb{R}, \tag{3}$$ is affine in $t$. We assume that the wave packet is normalized $$ 1~=~||\psi(t)||^2~=~ \langle \psi(t)| \psi(t)\rangle~=~\int_{\mathbb{R}} \!dp~A^2 .\tag{4}$$ The position operator in the momentum representation reads $$ \hat{x}~=~i\hbar\frac{\partial }{\partial p}.\tag{5}$$ Therefore the position average $$\langle \hat{x}\rangle~=~ \langle \psi(t)| \hat{x}|\psi(t)\rangle ~=~\hbar\int_{\mathbb{R}} \!dp~A^2\theta^{\prime} \tag{6}$$ is affine in $t$, and the average of the squared position $$ \langle \hat{x}^2\rangle~=~ \langle \psi(t)| \hat{x}^2|\psi(t)\rangle ~=~\hbar^2 \int_{\mathbb{R}} \!dp~(A^{\prime 2} +A^2\theta^{\prime 2} ) \tag{7} $$ is quadratic in $t$. Here primes denote differentiation wrt. $p$.
Thus we immediately know that the position variance $$ {\rm Var}(\hat{x})~=~at^2+bt+c\tag{8}$$ is quadratic in $t$ as well, where $a$, $b$, and $c$ are constant coefficients. It is straightforward to see from the Cauchy-Schwarz inequality $$\left(\int_{\mathbb{R}} \!dp~A^2\frac{p}{m} \right)^2 ~\leq~ \int_{\mathbb{R}} \!dp~\left(A\frac{p}{m}\right)^2 \cdot \int_{\mathbb{R}} \!dp~A^2 \tag{9}$$ that the second-order coefficient $$ a~=~\int_{\mathbb{R}} \!dp~\left(A\frac{p}{m}\right)^2-\left(\int_{\mathbb{R}} \!dp~A^2\frac{p}{m} \right)^2~\geq ~0, \tag{10}$$ is non-negative, cf. the normalization condition (4). In fact one may show that $a>0$ is strictly positive, and hence the ${\rm Var}(\hat{x})$ is a decreasing parabola for $t$ sufficiently negative, as we wanted to show.
Sketched indirect proof of $a\neq 0$: The second-order coefficient $a$ becomes zero $~\Leftrightarrow~$ the Cauchy-Schwarz inequality (9) becomes an equality $~\Leftrightarrow~$ $A(p) p$ is proportional to $A(p)$ $~\Leftrightarrow~$ $A(p)$ is proportional to a delta function $\delta(p-p_0)$. But this does not correspond to a normalizable wave packet.
Example. Two mutually approaching wave trains are an easy intuitive example where the position variance ${\rm Var}(\hat{x})$ diminishes in some time-interval $[t_1,t_2]$. But this is, in some sense, a lazy example, which sort of betrays just how universal and omnipresent this behavior is for quantum mechanics.
For instance, as we know from the general analysis in Section 2, already the simplest possible wave packet, i.e. a single Gaussian wave packet, displays this behavior. However, that is much less intuitive, and thus that much more fascinating/mind boggling to try to understand. Let us for simplicity set $\hbar=1=m$.
A single Gaussian wave packet at $t=0$ is of the form $$ \psi(p,t=0)~=~N \exp\left(-ipc - \frac{p^2}{2}\tau\right) ,\tag{11}$$ where $\tau>0$ and $c=a+ib\in\mathbb{C}$ are constants. Here $N>0$ is a normalization constant. The normalization condition (4) implies that $$ N~=~\left(\frac{\tau}{\pi}\right)^{\frac{1}{4}}\exp\left(-\frac{b^2}{2\tau}\right).\tag{12}$$ Thus for arbitrary time $t$, the Gaussian wave packet reads $$ \begin{align}\psi(p,t)~=~&\psi(p,0)\exp\left(-i\frac{p^2}{2}t\right)\cr ~=~& N \exp\left(-ipc - \frac{p^2}{2}(\tau+it)\right) .\end{align}\tag{13}$$ In the position representation, the Gaussian wave packet becomes $$ \begin{align}\psi(x,t) ~=~&\int_{\mathbb{R}} \frac{dp}{\sqrt{2\pi}} \exp\left(ipx\right) \psi(p,t)\cr ~=~&\frac{N}{\sqrt{\tau+it}} \exp\left(-\frac{(x-c)^2}{2(\tau+it)}\right).\end{align} \tag{14}$$ The position probability distributions becomes $$\begin{align}|\psi&(x,t)|^2\cr ~=~&\frac{N^2}{|\tau+it|} \exp\left(-{\rm Re}\frac{(x-c)^2}{\tau+it}\right) \cr ~=~&\frac{N^2}{\sqrt{\tau^2+t^2}} \exp\left(-\frac{[(x-a)^2-b^2]\tau-2(x-a)bt}{\tau^2+t^2}\right) \cr ~=~&\sqrt{\frac{\tau}{\pi(\tau^2+t^2)}} \exp\left(-\frac{(x-a-\frac{bt}{\tau})^2\tau }{\tau^2+t^2} \right). \end{align}\tag{15}$$ Therefore the position average $$ \langle \hat{x}\rangle~=~\int_{\mathbb{R}} \!dx~x|\psi(x,t)|^2 ~=~a+\frac{bt}{\tau}\tag{16}$$ is an affine function of $t$, while the position variance $$ {\rm Var}(\hat{x}) ~=~\int_{\mathbb{R}} \!dx~(x-\langle \hat{x}\rangle)^2|\psi(x,t)|^2 ~=~\frac{\tau^2+t^2}{2\tau}\tag{17}$$ is a quadratic function of $t$.
The time symmetric profile (17) of the position variance ${\rm Var}(\hat{x})$ of a single Gaussian wave packet is probably somewhat surprising to anyone who borrows his intuition from classical physics.
I don't know the answer to this yet, but here is a calculation that others might find useful in determining the answer. Let's compute the time derivative of $\sigma_X$. First note that $$ \frac{d}{dt}\sigma_X^2 = 2\sigma_X\dot\sigma_X, \qquad \sigma_X = \langle X^2\rangle - \langle X\rangle^2 $$ so \begin{align} \dot\sigma_X &= \frac{1}{2\sigma_X}\frac{d}{dt}\left(\langle X^2\rangle - \langle X\rangle^2\right) \\ &= \frac{1}{2\sigma_X}\left(\frac{d}{dt}\langle X^2\rangle - 2\langle X\rangle\frac{d}{dt}\langle X\rangle \right)\\ \end{align} Now use the general relation $$ \frac{d}{dt}\langle O\rangle = \frac{i}{\hbar}\langle[H,O]\rangle $$ for an operator with no explicit time-dependence. It follows that \begin{align} \frac{d}{dt}\langle X\rangle = \frac{i}{\hbar}\langle[P^2/2m, X]\rangle =\frac{i}{2m\hbar}\langle -2i\hbar P\rangle = \frac{\langle P\rangle}{m} \end{align} so that also \begin{align} \frac{d}{dt}\langle X^2\rangle &= \frac{i}{\hbar} \left\langle\left[\frac{P^2}{2m}, X^2\right]\right\rangle = \frac{i}{2m\hbar}(-2i\hbar)\langle\{P,X\}\rangle = \frac{\langle \{P,X\}\rangle}{m} \end{align} where $\{P, X\}=PX+XP$, and therefore $$ \dot\sigma_X = \frac{1}{2m\sigma_X}\Big(\langle \{P,X\}\rangle - 2\langle P \rangle \langle X \rangle\Big) $$ We want to know if there is a state for which at some time, $\dot\sigma_X<0$. Since $\sigma_X>0$, the expression just derived leads us to an equivalent question; is there a state for which the following inequality holds at some time? $$ \langle\{P,X\}\rangle < 2\langle P\rangle \langle X \rangle\qquad ? $$ Let me know if you find any errors in the math as I did it quickly.